Comonads and Convolutions

That scary Control.Comonad import from category-extras is going
to be the subject of today's post. We're going to look at one possible
implementation of comonads for non-empty lists that model causal
time-invariant systems, systems whose outputs depend only on inputs that
are in the past. We will see that computation in these systems follows
a comonadic structure and that one instance of this structure strongly
enforces causality and weakly enforces time-invariance.

Our causal lists are simply a newtype of list with the added
restriction that they are non-empty; causal is a "smart constructor"
that enforces this restriction. We use GeneralizedNewtypeDeriving
to get the Functor instance for free.

Background. (If you're already familiar with signal processing, feel
free to skip this section.) One such system models point-to-point
communication of voltage samples across an imperfect wire channel. In
an ideal world, we would very much like to be able to pretend that any
voltage I put into this channel would instantly perfectly transmit this
voltage to the other end of the channel. In practice, we'll see any
number of imperfections, including time to rise and fall, a delay, ringing and noise. Noise is a party pooper, so
we're going to ignore it for the purposes of this post.

To a first approximation, we can impose the following important
conditions on our system:

Causality. Our wire can't peek into the future and transmit
some voltage before it has even gotten it.

Time-invariance. Any signal will get the same response whether or
not it gets sent now or later.

Linearity. A simple and useful approximation for wires, which
states this mathematical property: if an input x1 results in an
output y1, and an input x2 results in an output y2, then the
input Ax1 + Bx2 results in the output Ay1 + By2. This also means
we get superposition, which is an important technique that we'll use
soon.

When you see a linear time-invariant system, it means that we get to use
a favorite mathematical tool, the convolution.

Discrete convolutions. The overall structure of the discretized
computation that a channel performs is [Voltage] -> [Voltage]; that
is, we put in a sequence of input voltage samples, and get out another
sequence of output voltage samples. On the other hand, the discrete
convolution is the function calculated by (with variables suggestively
named):

(u ∗ f)[n] = sum from m = -∞ to ∞ of f[m]u[n-m]

It's not quite obvious why the convolution is the mathematical
abstraction we're looking for here, so we'll sketch a brief derivation.

One special case of our computation is when the input corresponds to
[1, 0, 0, 0 ...], called the unit sample. In fact, due to
linearity and time-invariance, the output that our system gives when
posed with the unit sample, the unit sample response, precisely
specifies the behavior of a system for all inputs: any possible input
sequence could be composed of any number of delayed and scaled unit
samples, and linearity says we can sum all of the results together to
get a result.

A list is actually a function ℕ → a, and we can extend the domain to
be over integers if we propose the convention f[n] = 0 for n <
0. Suppose that f[n] represents our input samples varying over
time, δ[n] represents a unit sample (δ[0] = 1, δ[n] = 0 for
all other n; you'll commonly see δ[n-t], which is a unit sample
at time t), and u[n] represents our unit sample response. Then,
we decompose f[n] into a series of unit samples:

which looks just like the discrete convolution, just without the -∞
bound. Remember that we defined f[m] = 0 for m < 0, so the
two are actually equivalent.

I'd like to linger on that final mathematical definition for a moment,
before writing out the equivalent Haskell. We originally stated that
the input-response computation had the type [Voltage] -> [Voltage];
however, in our math, we've actually defined a relation [Voltage] ->
Voltage, a channel specific function that takes all of the inputs up
to time n, i.e. f[0]..f[n], and returns a single output
g[n]. I've written the following definition in a suggestive curried
form to reflect this:

The unit sample response may be a finite or infinite list, for
reasons of efficiency a finite list is recommended:

> usr :: [Voltage]
> usr = [1,2,5,2,1]

Comonads. By now, it should be clear where we've been working
towards: we have ltiChannel usr :: Causal Voltage -> Voltage and we
want: Causal Voltage -> Causal Voltage. This is precisely the form
of computation that the comonad induces! For your convenience, here is
the definition of the Copointed and Comonad type classes:

The Copointed instance is straight-forward, but demonstrates why the
Causal must contain a non-empty list:

> instance Copointed Causal where
> extract (Causal xs) = head xs

The Comonad instance can be defined using either duplicate or
extend; both have default implementations defined in terms of each
other. Deriving these default implementations is left as an exercise to
the reader; we'll define both here:

The function duplicate really gets to the heart of what this comonad
instance does: we take our input list and transform it into a list of
histories, each one one step further than the last. The tail tags
along to drop the first value of inits which is an empty list.
duplicate builds up w (w a), and then the user-supplied function
tears it back down to w b (if you think of monads, the lifted user
function builds up m (m b), and then join tears it back down to
m b.)

=>> is a flipped extend, and the comonadic equivalent of the
monadic >>=.

Enforced invariants. Structuring our computation in this form (as
opposed to writing the darn convolution out explicitly) gives us some
interesting enforced invariants in our code. Our channels need not be
linear; I could have squared all of the inputs before convolving them
with the unit sample response, and that certainly would not be linear.
However, any channel we write must be causal and and will usually be
time-invariant: it must be causal because we never pass any values from
the future to the user function, and it is weakly time invariant because
we don't explicitly let the user know how far along the are the input
stream they are. In practice with our implementation, they could divine
this information using length; we could get stronger guarantees
employing a combinator that reverses the list and then appends repeat
0:

u in this case must be finite, and if it is infinite can be
truncated at some point to specify how precise our computation should
be.

Open question. The unit sample response has been expressed in our
sample code as [Voltage], but it really is Causal Voltage.
Unfortunately, the comonad doesn't seem to specify mechanisms for
combining comonadic values the same way the list monad automatically
combines the results of computations for each of the values of a list.
I'm kind of curious how something like that might work.

5 Responses to “Comonads and Convolutions”

In order to combine such values, you need to support the Applicative interface as well, or at least provide a zipping facility to pair up streams. See the paper The Essence of Dataflow Programming (by Uustalu and Vene) for details.